A space-time variational approach to hydrodynamic stability theory

We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space–time variational formulation and associated generalized singular value decomposition of the (linearized) Navier–Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary- or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a ‘data’ space–time norm; the theory provides a rigorous, sharp (realizable) and efficiently computed bound for the velocity perturbation measured in a ‘solution’ space–time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space–time semi-norms; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi–Rappaz–Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds number flow in an eddy-promoter channel.United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550-09-1-0613)United States. Office of Naval Research (Grant N00014-11-1-0713

On Dual-Weighted Residual Error Estimates for p-Dependent Discretizations

This report analyzes the behavior of three variants of the dual-weighted residual (DWR) error estimates applied to the p-dependent discretization that results from the BR2 discretization of a second-order PDE. Three error estimates are assessed using two metrics: local effectivities and global effectivity. A priori error analysis is carried out to study the convergence behavior of the local and global effectivities of the three estimates. Numerical results verify the a priori error analysis

Massively Parallel Solver for the High-Order Galerkin Least-Squares Method

A high-order Galerkin Least-Squares (GLS) finite element discretization is combined with massively parallel implicit solvers. The stabilization parameter of the GLS discretization is modified to improve the resolution characteristics and the condition number for the high-order interpolation. The Balancing Domain Decomposition by Constraints (BDDC) algorithm is applied to the linear systems arising from the two-dimensional, high-order discretization of the Poisson equation, the advectiondiffusion equation, and the Euler equation. The Robin-Robin interface condition is extended to the Euler equation using the entropy-symmetrized variables. The BDDC method maintains scalability for the high-order discretization for the diffusiondominated flows. The Robin-Robin interface condition improves the performance of the method significantly for the advection-diffusion equation and the Euler equation. The BDDC method based on the inexact local solvers with incomplete factorization maintains the scalability of the exact counterpart with a proper reordering.This work was partially supported by funding from The Boeing Company with technical monitor of Dr. Mori Mani

The generalized Empirical Interpolation Method: stability theory on Hilbert spaces with an application to the Stokes equation

International audienceThe Generalized Empirical Interpolation Method (GEIM) is an extension first presented in [1] of the classical empirical interpolation method (see [2], [3], [4]) where the evaluation at interpolating points is replaced by the evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in [1], this allows to relax the continuity constraint in the target functions and expand the application domain. A special effort has been made in this paper to understand the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it in the first part of the paper to an inf-sup problem in the case of Hilbert spaces. In the second part, it will be explained how GEIM can be employed to monitor in real time physical experiments by combining the acquisition of measurements from the processes with their mathematical models (parameter-dependent PDE's). This idea will be illustrated through a parameter dependent Stokes problem in which it will be shown that the pressure and velocity fields can efficiently be reconstructed with a relatively low dimension of the interpolating spaces

Microsurgery training using Apple iPad Pro

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/146635/1/micr30384.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/146635/2/micr30384_am.pd

Accelerated nonlinear PDE-constrained optimization by reduced order modelling

We present a framework to accelerate optimization of problems where the objective function is governed by a nonlinear partial differential equation (PDE) using projection-based reduced-order models (ROMs) and a trust-region (TR) method. To reduce the cost of objective function evaluations by several orders of magnitude, we replace the underlying full-order model (FOM) with a series of hyperreduced ROMs (HROMs) constructed on-the-fly. Each HROM is equipped with an online-efficient a posteriori error estimator, which is used to define a TR. Hyperreduction is performed following a goal-oriented empirical quadrature procedure, which guarantees first-order consistency of the HROM with the FOM at the TR center. This ensures the optimizer converges to a local minimum of the underlying FOM problem. We demonstrate the framework through optimization of a nonlinear thermal fin and pressure-matching inverse design of an airfoil under Euler flow and Reynolds-averaged Navier-Stokes flow

A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics

We present a parameterized-background data-weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; weak greedy construction of prior (background) spaces associated with an underlying potentially high-dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations; and finally, output prediction from the optimality saddle in O(M[superscript 3) operations, where M is the number of experimental observations. We present results for a synthetic Helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. To conclude, we consider a physical raised-box acoustic resonator chamber: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the time-harmonic pressure field; we demonstrate the considerable improvement in prediction provided by the integration of a best-knowledge model and experimental observations; we extract, even from these results with real data, the numerical trends indicated by the theoretical convergence and stability analyses.Fondation Sciences Mathematiques de ParisUnited States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550-09-1-0613)United States. Office of Naval Research (Grant N00014-11-1-0713)SUTD-MIT International Design Centr

An LP empirical quadrature procedure for parametrized functions

We extend the linear program empirical quadrature procedure proposed in and subsequently to the case in which the functions to be integrated are associated with a parametric manifold. We pose a discretized linear semi-infinite program: we minimize as objective the sum of the (positive) quadrature weights, an ℓ[subscript 1] norm that yields sparse solutions and furthermore ensures stability; we require as inequality constraints that the integrals of J functions sampled from the parametric manifold are evaluated to accuracy [¯ over δ]. We provide an a priori error estimate and numerical results that demonstrate that under suitable regularity conditions, the integral of any function from the parametric manifold is evaluated by the empirical quadrature rule to accuracy [¯ over δ] as J→∞. We present two numerical examples: an inverse Laplace transform; reduced-basis treatment of a nonlinear partial differential equation.United States. Office of Naval Research (Grant N00014-17-1-2077